$$\beta $$-Ensembles and higher genera Catalan numbers

Abstract

We propose formulas for the large N expansion of the generating function of connected correlators of the $$\beta $$ β -deformed Gaussian and Wishart–Laguerre matrix models. We show that our proposal satisfies the known transformation properties under the exchange of $$\beta $$ β with $$1/\beta $$ 1 / β and, using Virasoro constraints, we derive a recursion formula for the coefficients of the expansion. In the undeformed limit $$\beta =1$$ β = 1 , these coefficients are integers and they have the combinatorial interpretation of generalized Catalan numbers. For generic $$\beta $$ β , we define the higher genus Catalan polynomials $$C_{g,\nu }(\beta )$$ C g , ν ( β ) whose coefficients are integer numbers.